reserve x,X,X2,Y,Y2 for set;
reserve GX for non empty TopSpace;
reserve A2,B2 for Subset of GX;
reserve B for Subset of GX;

theorem
  for A, V be Subset of GX st A is a_component & V is connected &
  V c= A & V<>{} holds A = Component_of V
proof
  let A, V be Subset of GX;
  assume that
A1: A is a_component and
A2: V is connected and
A3: V c= A and
A4: V<>{};
  V c= Component_of V by A2,Th1;
  then
A5: A meets (Component_of V) by A3,A4,XBOOLE_1:67;
  assume
A6: A <> Component_of V;
  Component_of V is a_component by A2,A4,Th8;
  hence contradiction by A1,A6,A5,CONNSP_1:1,34;
end;
